PART 2b. PLANAR KINETICS OF LUMPED MASS SYSTEMS
Motion and
Deformation of Mechanical Systems with 2 Degrees of Freedom (2DOF)
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Textbook Chapter 3.5 (Lectures 1418)
Acronyms: M:
mass, K: stiffness, C: damping, ODE: ordinary differential equation, EOM:
equation of motion, SEP: static equilibrium position, DOF: degree of freedom,
FBD: free body diagram, CE: characteristic equation, CME: Principle of
Conservation of Mechanical Energy
Lecture
(get me)

Major Topics/WHAT
YOU WILL LEARN

Recommended homework
problems

14

EOMs for 2DOF mechanical MKC systems including
base excitation. Free body diagrams (FBDs),
selection of coordinates, and establishment of forces for mechanical elements
connecting masses. Application of Newton’s
2^{nd} Law. Expressing EOMS in matrix form. Derivation of nonlinear
EOMs for double pendulum.

3.49,
3.48, 3.50

15

Eigenanalysis of free response of 2DOF undamped
mechanical systems. Fundamental response function and derivation of system
characteristic equation (CE). Solving CE: Eigenvalues (natural frequencies)
and eigenvectors (mode shapes) of system, physical interpretation of natural
frequencies and mode shapes. Transformation of coordinates to modal space via
modal matrix of eigenvectors and uncoupling of system EOMs.

3.52ad,
3.53ad, 3.54

16

Transient response of 2DOF undamped mechanical
systems. Prediction of system transient (free) response using modal (natural)
coordinates. Transformation to physical space. Examples of analysis and
interpreting solutions (system responses). Examples of systems with rigid
body motions: interpretation of null or zero natural frequency. Insights into
the analysis of systems with viscous damping: lightly damped systems and systems
with proportional damping. The concept of modal damping ratio. Application to
the analysis of a 2DOF system response after collision

3.52ef,
3.53ef, 3.56

17

Transient response of 2 DOF MKC system
with proportional damping. Example of usage of modal coordinates – Collision
problem

3.60, 3.61

18

Forced periodic response of
2DOF undamped mechanical systems. Steady state solution to harmonic force
excitation using (a) modal coordinates and (b) direct substitution.
Interpretation of regimes of operation: below, around and above natural
frequencies. Effect of excitation frequency on amplitude (amplification
factor) and phase lag of steadystate response. Insights into the forced
periodic response of 2DOF systems with viscous damping

3.62, 3.63 (forget the transient solution due to initial conditions
and solve for the steadystate solution).
Also, repeat with modal damping ratios = 0.2

Appendix

Application: The vibration Absorber


Get all: Lectures
1418_{}
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